File size: 3,358 Bytes
d4317d7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import OAI.MeasureTheory.Falconer.Geometry.SeparatedProjectionBases
import OAI.MeasureTheory.Falconer.Incidence.FiniteBallBounds
import OAI.MeasureTheory.Falconer.Geometry.ProjectionCells
namespace OAI
/-! The two bases and every graph slice are selected from the original
common projection family, retaining explicit relative mass and regularity. -/
namespace Falconer
attribute [local instance] Classical.propDecidable
theorem common_projection_configuration {δ t K ρ r : ℝ}
(hδ : 0 ≤ δ) (hK : 0 ≤ K) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (hr : 0 < r)
(I : Finset ℝ) (U : Finset (Space 2)) (A : ℝ → Finset (Space 2))
(hI : I.Nonempty) (hU : U.Nonempty)
(hsub : ∀ i ∈ I, A i ⊆ U)
(hsize : ∀ i ∈ I, ρ*(U.card:ℝ) ≤ ((A i).card:ℝ))
(hball : ∀ z : ℝ, scalarBallMass I z r ≤ (ρ^6/128)*(I.card:ℝ))
(hUreg : NormalizedFiniteBallBound δ t K U)
(P : ℕ) (hproj : ∀ i ∈ I, (projectionCells δ i (A i)).card ≤ P) :
∃ a ∈ I, ∃ b ∈ I, r < |b-a| ∧ b ≠ a ∧ ∃ J ⊆ I,
J.Nonempty ∧ (ρ^6/64)*(I.card:ℝ) ≤ (J.card:ℝ) ∧
let V := A a ∩ A b
V.Nonempty ∧ V ⊆ U ∧ (U.card:ℝ) ≤ (8/ρ^4)*(V.card:ℝ) ∧
NormalizedFiniteBallBound δ t (K*(8/ρ^4)) V ∧
(projectionCells δ a V).card ≤ P ∧ (projectionCells δ b V).card ≤ P ∧
∀ j ∈ J, r < |j-a| ∧ r < |j-b| ∧
(ρ^4/8)*(V.card:ℝ) ≤ ((V ∩ A j).card:ℝ) ∧
(projectionCells δ j (V ∩ A j)).card ≤ P := by
classical
obtain ⟨a,ha,b,hb,hab,hba,J,hJI,hJsize,htriple⟩ :=
separated_projection_bases I U A hI hU hρ hρ1 hr hsub hsize hball
have hJ : J.Nonempty := by
apply Finset.card_pos.mp
have hp : 0 < (ρ^6/64)*(I.card:ℝ) :=
mul_pos (by positivity) (Nat.cast_pos.mpr hI.card_pos)
exact_mod_cast hp.trans_le hJsize
let V := A a ∩ A b
have hVU : V ⊆ U := Finset.inter_subset_left.trans (hsub a ha)
obtain ⟨j,hj⟩ := hJ
have hVlower : (ρ^4/8)*(U.card:ℝ) ≤ (V.card:ℝ) :=
(htriple j hj).2.2.trans (Nat.cast_le.mpr (Finset.card_le_card Finset.inter_subset_left))
have hV : V.Nonempty := by
apply Finset.card_pos.mp
have hp : 0 < (ρ^4/8)*(U.card:ℝ) :=
mul_pos (by positivity) (Nat.cast_pos.mpr hU.card_pos)
exact_mod_cast hp.trans_le hVlower
have hVUsize : (U.card:ℝ) ≤ (8/ρ^4)*(V.card:ℝ) := by
have hpos : 0 < ρ^4 := by positivity
have hh := mul_le_mul_of_nonneg_left hVlower (show 0 ≤ 8/ρ^4 by positivity)
have he : (8/ρ^4)*((ρ^4/8)*(U.card:ℝ)) = (U.card:ℝ) := by field_simp
rwa [he] at hh
refine ⟨a,ha,b,hb,hab,hba,J,hJI,⟨j,hj⟩,hJsize,hV,hVU,hVUsize,
normalized_finite_ball_bound_subset hδ hK hVU hVUsize hUreg,?_,?_,?_⟩
· exact (Finset.card_le_card (projectionCells_mono δ a Finset.inter_subset_left)).trans (hproj a ha)
· exact (Finset.card_le_card (projectionCells_mono δ b Finset.inter_subset_right)).trans (hproj b hb)
· intro j hj
obtain ⟨hja,hjb,hcard⟩ := htriple j hj
refine ⟨hja,hjb,?_,?_⟩
· exact (mul_le_mul_of_nonneg_left (Nat.cast_le.mpr (Finset.card_le_card hVU))
(show 0 ≤ ρ^4/8 by positivity)).trans hcard
· exact (Finset.card_le_card (projectionCells_mono δ j Finset.inter_subset_right)).trans (hproj j (hJI hj))
end Falconer
end OAI
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